Remark 5.5 It was shown in [1]} that if \((G\rightrightarrows M,\eta )\)  is a Riemannian groupoid (see Section REF ) and \(\mathcal {O}\subseteq M\)  is any orbit, then the exponential map defines a weak linearization of \(G\)  around \(\mathcal {O}\) . Also, a result of [1]} establishes that any proper Lie groupoid admits a 2-metric, hence if \(G\rightrightarrows M\)  is proper groupoid, then it is weakly linearizable around \(\mathcal {O}\) . The proof of the Morse lemma we exhibited in Theorem REF  reveals the geometric richness derived from the existence of an Euler-like multiplicative vector field. Nevertheless, a less simple proof of such a result may be given by assuming the weakly linearization provided by a 2-metric combined with the classical ideas for the proof of the Morse-Bott lemma that can be found for instance in [3]}. The latter approach has the advantage of not requiring our Lie groupoid to be proper.

